#852768

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(2,3)(4,8)(10,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,22)(19,24)(26,28)(30,31);(17,19)(18,20)(21,24)(22,23)(25,27)(26,30)(28,31)(29,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,20)(21,24)(22,23)(25,28,29,26)(27,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=gi,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336761, #72336764, #72336875
maximal normal subgroup of	#72336761, #72336764, #72336875

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 759964);